Noam M. D. Kolodner | Searcharxiv

arXiv2022

A representation of $\text{Out}\left(F_{n}\right)$ by counting subwords of cyclic words

We generalize the combinatorial approaches of Rapaport and Higgins--Lyndon to the Whitehead algorithm. We show that for every automorphism $φ$ of a free group $F$ and every word $u\in F$ there exists a finite multiset of words $S_{u,φ}$ satisfying the following property: For every cyclic word $w$, the number of times $u$ appears as a subword of $φ\left(w\right)$ depends only on the appearances of words in $S_{u,φ}$ as subwords of $w$. We use this fact to construct a faithful representation of $\text{Out}\left(F_{n}\right)$ on an inverse limit of $\mathbb{Z}$-modules, so that each automorphism is represented by sequence of finite rectangular matrices, which can be seen as successively better approximations of the automorphism.

math.GR↗

arXiv2020

Classes of free group extensions

In this paper we identify different classes of free group extension using core graphs. We show that every free group extension $H\leq K\leq F$ has a base $B$ such that the associated pointed graph morphism $Γ_{B}\left(H\right)\toΓ_{B}\left(H\right)$ is onto. But if we examine graphs without base points, there is an extension $\left\langle b\right\rangle \leq\left\langle b,aba^{-1}\right\rangle <F_{\left\{ a,b\right\} }$ such that for every base of $F_{\left\{ a,b\right\} }$ the associated graph morphisms are injective.

math.GR↗